Those booked on flights in the next few weeks face uncertainty after staff at BAA's six airports, including Heathrow, voted to take industrial action in a row over pay.
But travellers were offered a glimmer of hope after conciliation service Acas said it was arranging a meeting between BAA and the workers' union Unite.
Unite said no decision over the form or date of any strike will be taken before Monday which would leave the weekend open for talks.
The earliest a strike could be called would be Monday August 23 - just ahead of the busy bank holiday weekend.
With the Unite members including firefighters and security staff, any walkout would have a devastating effect on UK air travel which was already hit by a separate dispute involving British Airways' cabin crew, also represented by Unite.
Despite only half of the 6,185 Unite staff voting in the ballot, its leadership said BAA faced "a total shutdown" of its six airports - Heathrow, Stansted, Southampton, Glasgow, Edinburgh and Aberdeen. Of those who did vote, almost three quarters (74.1%) came out in favour of strike action.
BAA said it did not believe the result provided "a clear mandate for strike action".
Prime Minister David Cameron said a strike would "do nothing but harm" and would damage business, jobs and tourism. Transport Secretary Philip Hammond urged both sides to seek an "amicable solution".
Unite national officer Brendan Gold and the union's national officer for civil aviation, Brian Boyd said the union would meet its shop stewards on Monday to decide what form of industrial action members would take.
Editor's note;
BAA need to start talking to the RMT, because even those they do "recognise" don't seem too impressed with the treatment their members suffer at the hands of BAA.
If a picket line is created as part of this impending industrial action, will Unite taxi branch members be prepared to cross it?
